A necessary and sufficient condition for all the eigenvalues of a real matrix to have negative real parts is that the equation

has as a solution where is an matrix and is a positive definite quadratic form.

A necessary and sufficient condition for all the eigenvalues of a real matrix to have negative real parts is that the equation

has as a solution where is an matrix and is a positive definite quadratic form.

Weisstein, Eric W. "Lyapunov's First Theorem."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/LyapunovsFirstTheorem.html